A simple random sample of size n = 12 is obtained from a population with μ = 63 and σ = 19.

(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of x̅.

(b) Assuming the normal model can be used, determine P(x̅ < 66.9).

(c) Assuming the normal model can be used, determine P(x̅ ≤ 65.3).

(d) Compare the results obtained in parts (b) and (c) with the results obtained in parts (b) and (c) in Problem 17.

What effect does increasing the sample size have on the probabilities? Why do you think this is the case?

Data from Problem 17

A simple random sample of size n = 12 is obtained from a population with μ = 69 and σ = 15.

What must be true regarding the distribution of the population in order to use the normal model to compute probabilities regarding the sample mean? Assuming the normal model can be used, describe the sampling distribution x.

Assuming the normal model can be used, determine P(x̅ < 72.6).

Assuming the normal model can be used, determine P(x̅ ≥ 70.2).